Problem | #1 | Extra credit |
#2 | #3 | #4 | #5 | #6 | #7 | Total |
---|---|---|---|---|---|---|---|---|---|

Max grade | 15 | 3 | 14 | 14 | 12 | 14 | 15 | 15 | 94 |

Min grade | 5 | 0 | 0 | 1 | 1 | 2 | 2 | 0 | 35 |

Mean grade | 12.28 | 1.81 | 8.81 | 7.5 | 8.5 | 7.31 | 8.25 | 11.69 | 66.16 |

Median grade | 14 | 3 | 9.5 | 8 | 9 | 6 | 7.5 | 13 | 67.25 |

Numerical grades will be retained for use in computing the final letter grade in the course. Here are approximate letter grade assignments for this exam:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

Range | [85,100] | [80,84] | [70,79] | [65,69] | [55,64] | [50,54] | [0,49] |

**Problem 1 (15 points)**

2 points for a sketch of the contour.

6 points for the residue
computation.

6 points for showing that the ML estimate makes the
integral over the "other" part of the contour go to 0.

1 point for getting the answer.

3 points for the correct extra credit answer.

**Problem 2 (14 points)**

a) 7 points. simply and simple each get 1 point, properly used. The
equation itself gets 3 points. The other "stuff" gets 2 points.

b) 7 points. The answer, and some method for getting it.

**Problem 3 (15 points)**

The singularity at 0 can earn 7 points. The other singularities can
earn 8 points.

At 0, 1 point for the statement that the singularity is removable and
1 point for the value of the residue. The other 5 points are for some
supporting evidence.

Similar scoring will be used for the
non-zero singularities: 1 point for the residue and 1 point for
identifying the singularity as a pole and 1 point for giving the order
of the pole. Again, supporting evidence can earn 5 points.

**Problem 4 (12 points)**

a) 8 points. Apply Liouville's Theorem (2 points) to a "correct"
function (4 points) and get the conclusion (2 points).

b) 4 points. 2 for the assertion that the exponential function does
not have modulus bounded away from 0. 2 more points for explaining
why.

**Problem 5 (14 points)**

2 points for information about a power series for sin(z).

3 points for information about a power series for
1/(z-1)^{2}.

5 points for combining them usefully (multiplication, division,
etc.)

Then 4 points for each of the correct terms in the answer.

An alternative unrecommended direct approach is possible. So computing
correctly f_{(k)(0) will earn k points, where k is an integer
running from 1 to 4. Thus computing f(0) itself earns nothing
(sigh). Assembling the terms in the Taylor series (with the
factorials) earns, as above, 4 points for each of the correct terms in
the answer.
}

**Problem 6 (14 points)**

a) 4 points. Many correct answers are possible, and if another answer
(inferentially) needs to be chosen to answer d), that's o.k. 2 points
for excluding 1, and 2 points for making a correct "cut" in **C**.
b) 2 points for the correct answer.

c) 2 points for the correct answer.

d) 7 points. Some derivatives of sqrt(z) will earn 2 points. This
should be a series in integer powers of z-1. Each correct term will
earn 1 point.

**Problem 7 (15 points)**

5 points for instantiating the conventional "dictionary" changing this
to a complex line integral. Finding the singularities of the integrand
and manipulating it algebraically correctly earns another 5
points. Applying the Residue Theorem, and computing the correct
residue and getting the correct answer earns the final 5 points. The
not-uncommon error of failing to compensate for the fact that the
dictionary response does not get a monic polynomial will lose 2 points
(this usually yields an extra b in the answer).

**
Maintained by
greenfie@math.rutgers.edu and last modified 4/21/2005.
**